<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #);<br/>
<a NAME="E1:37_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) is <a href="rlvect_1.html#V3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<a NAME="E1:37_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>
<a NAME="E4:37_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:37_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:37_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:37_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectsp10.html#E6:37_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E10:37_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectsp10.html#E6:37_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E10:37_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E2:37_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) is <a href="rlvect_1.html#V4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><br/>

<a NAME="E1:37_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>
<a NAME="E4:37_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:37_1_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>
<a NAME="E7:37_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E9:37_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:37_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:37_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E15:37_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<a NAME="E16:37_1_1_2"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp10.html#E6:37_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E11:37_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp10.html#E13:37_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E17:37_1_1_2"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="vectsp10.html#E6:37_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E9:37_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E13:37_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="vectsp10.html#E15:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E18:37_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<a NAME="E19:37_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E9:37_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E15:37_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="vectsp10.html#E16:37_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp10.html#E11:37_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp10.html#E17:37_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp10.html#E19:37_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E3:37_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) is <a href="rlvect_1.html#V5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<a NAME="E1:37_1_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<i><font color="Green">E27</font></i>:  <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) = 
the <a href="struct_0.html#U2">Zero</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #)

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T60">VECTSP_4:60</a></i>
;<br/>
<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #)</span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #)</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E5:37_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E6:37_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="vectsp10.html#E5:37_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:37_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<a NAME="E1:37_1_1_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:37_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<i><font color="Green">E28</font></i>:  <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) = 
the <a href="struct_0.html#U2">Zero</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #)

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_4.html#T60">VECTSP_4:60</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E7:37_1_1_4"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/>
<a NAME="E8:37_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) ;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E5:37_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #)
<b>by </b><i><a class="txt" href="vectsp10.html#E6:37_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #);<br/><a NAME="E1:37_1_1_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:37_1_1_5"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E4:37_1_1_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:37_1_1_5"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:37_1_1_5"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:37_1_1_5"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><a NAME="E11:37_1_1_5"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E6:37_1_1_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="txt" href="vectsp10.html#E10:37_1_1_5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E12:37_1_1_5"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:37_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><a NAME="E14:37_1_1_5"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E15:37_1_1_5"/><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>;<br/><a NAME="E16:37_1_1_5"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E6:37_1_1_5"><i><font color="Green">E27</font></i></a>, <a class="txt" href="vectsp10.html#E10:37_1_1_5"><i><font color="Green">E29</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E17:37_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><a NAME="E18:37_1_1_5"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
the <a href="vectsp_1.html#U2">lmult</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E11:37_1_1_5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E12:37_1_1_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp10.html#E14:37_1_1_5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp10.html#E16:37_1_1_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="vectsp10.html#E18:37_1_1_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E15:37_1_1_5"><i><font color="Green">E33</font></i></a></i>
;<br/><a NAME="E20:37_1_1_5"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E21:37_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> is    <a href="vectsp_4.html#M2">Coset</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D6">VECTSP_4:def 6</a></i>;<br/><a NAME="E22:37_1_1_5"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E23:37_1_1_5"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E12:37_1_1_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="vectsp10.html#E14:37_1_1_5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp10.html#E20:37_1_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp10.html#E22:37_1_1_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="vectsp10.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E15:37_1_1_5"><i><font color="Green">E33</font></i></a>, <a class="txt" href="vectsp10.html#E23:37_1_1_5"><i><font color="Green">E38</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E20:37_1_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="vectsp10.html#E23:37_1_1_5"><i><font color="Green">E38</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="vectsp_4.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp10.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:37_1_1_5"><i><font color="Green">E26</font></i></a>, <a class="txt" href="vectsp10.html#E8:37_1_1_5"><i><font color="Green">E28</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# <span class="p1">(<span class="default"><a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#V12">VectSp-like</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="vectsp10.html#E1:37_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="vectsp10.html#E2:37_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="vectsp10.html#E3:37_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="vectsp10.html#E4:37_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E7:37_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp10.html#K2" target="_self">CosetSet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="vectsp10.html#K3" target="_self">addCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp10.html#K4" target="_self">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="vectsp10.html#K5" target="_self">lmultCoset</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


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